# Tetraoctagonal tiling

In geometry, the tetraoctagonal tiling is a uniform tiling of the hyperbolic plane.

Tetraoctagonal tiling

Poincaré disk model of the hyperbolic plane
Type Hyperbolic uniform tiling
Vertex configuration (4.8)2
Schläfli symbol r{8,4} or ${\displaystyle {\begin{Bmatrix}8\\4\end{Bmatrix}}}$
rr{8,8}
rr(4,4,4)
t0,1,2,3(∞,4,∞,4)
Wythoff symbol 2 | 8 4
Coxeter diagram or
or

Symmetry group [8,4], (*842)
[8,8], (*882)
[(4,4,4)], (*444)
[(∞,4,∞,4)], (*4242)
Dual Order-8-4 quasiregular rhombic tiling
Properties Vertex-transitive edge-transitive

## Constructions

There are for uniform constructions of this tiling, three of them as constructed by mirror removal from the [8,4] or (*842) orbifold symmetry. Removing the mirror between the order 2 and 4 points, [8,4,1+], gives [8,8], (*882). Removing the mirror between the order 2 and 8 points, [1+,8,4], gives [(4,4,4)], (*444). Removing both mirrors, [1+,8,4,1+], leaves a rectangular fundamental domain, [(∞,4,∞,4)], (*4242).

Name Image Symmetry Schläfli Tetra-octagonal tiling Rhombi-octaoctagonal tiling [8,4](*842) [8,8] = [8,4,1+](*882) = [(4,4,4)] = [1+,8,4](*444) = [(∞,4,∞,4)] = [1+,8,4,1+](*4242) = or r{8,4} rr{8,8}=r{8,4}1/2 r(4,4,4)=r{4,8}1/2 t0,1,2,3(∞,4,∞,4)=r{8,4}1/4 = = = or

## Symmetry

The dual tiling has face configuration V4.8.4.8, and represents the fundamental domains of a quadrilateral kaleidoscope, orbifold (*4242), shown here. Adding a 2-fold gyration point at the center of each rhombi defines a (2*42) orbifold.